The \(k\)th prime is greater than \(k(\ln k+\ln\ln k-1)\) for \(k\geq 2\). (English) Zbl 0913.11039

Dusart, Pierre

Math. Comput. 68, No. 225, 411-415 (1999).

The prime number theorem shows that the \(k\)-th prime \(p_k\) has the asymptotic value \(k\log k\) as \(k\to\infty\). Applying the theorem with an error term, M. Cipolla [Napoli Rend (3) 8, 132–166 (1902; JFM 33.0214.04)] obtained an asymptotic formula for \(p_k\) with leading terms \(k(\log k+\log\log k-1+\cdots)\) and an error term \(O\big(k(\log\log k/\log k)^3\big)\).
Good explicit bounds for \(p_k\) can be obtained from estimates for the Chebyshev functions and calculations of the zeros of the Riemann-zeta function. Thus, J. B. Rosser [Proc. Lond. Math. Soc. (2) 45, 21–44 (1939; Zbl 0019.39401)] first found that \(p_k>k\log k\) for \(k>1\), and together with L. Schoenfeld [Math. Comput. 29, 243–269 (1975; Zbl 0295.10036)] extended the result to \(p_k>k(\log k+\log\log k-c)\), with \(c=3/2\). Later, G. Robin [Acta Arith. 42, 367–389 (1983; Zbl 0525.10024)] improved this to \(c=1.0072629\), and together with J.-P. Massias [J. Théor. Nombres Bordx. 8, 215–242 (1996; Zbl 0856.11043)] found that \(c=1\) is admissible for \(1<k\leq\exp(598)\) and \(k\geq\exp(1800)\).
The author shows that the estimate \[ | \psi(x)-x| \leq 0.905{\times}10^{-7}x \] holds for \(x\geq\exp(50)\), thereby deducing that the above result with \(c=1\) is valid for all \(k>1\). It is also shown that \[ k(\log p_k-2)<p_k<k\min(\log p_k,\log k+\log\log k)\quad\text{when } k\geq 6. \]

Reviewer: Peter Shiu (Loughborough)

MSC:

11N05	Distribution of primes
11A41	Primes
11N37	Asymptotic results on arithmetic functions
11N56	Rate of growth of arithmetic functions

Keywords:

distribution of primes; arithmetic functions; explicit extimates; asymptotic value

Citations:

Zbl 0019.39401; Zbl 0295.10036; Zbl 0525.10024; Zbl 0856.11043; JFM 33.0214.04

References:

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